Does it make sense for a single driver to own 2 cars? Debating whether to trade in my old car when I buy another or keep it

I'm an owner of a 2000 Honda Civic EX with approximately 241,700 miles on the odometer. I've had the car since 2005 and have recently decided to upgrade to a newer vehicle and keep the Civic as an extra commuter/beater car for local travel. After taking the Civic to the shop due to a clunking noise coming from the front of the car while driving, it was found out that the tie rods are pretty much shot. The entire rack & pinion assembly will need to be replaced as well as it has been leaking for some time, but it's been a slow, manageable leak. Other issues found by the mechanic include worn, leaking front struts and a leaking radiator, which will need to be replaced sometime soon, but can probably wait a bit longer. However, the engine, clutch, transmission, air conditioning tires and brakes are all still in good shape.
My question is sort of a two part question: 1) is this car worth keeping given its condition? 2) Does it make sense to own a second vehicle when I'm the only person that will use both? I do frequent short trips where I live and I figure it might make sense to put most of that mileage on the older car. The estimated cost for the repairs is about $1300.
So am I crazy to keep this car? Or should I just let it go and take the $500 the dealer is offering? I guess a big part of me doesn't want to let it go since I've had it for so long and I've always figured I'd run the car to the ground before getting rid of it.
I appreciate any advice or insight.

I wouldnt use it for a trade in, i would sell it outright and use that money for the down pmt. You'll get more out of it if you sell it to a individual. You should be able to get a thousand out of it easy.

I could see keeping a second car if you need an old beater pick-up for things like DIY home improvement projects, camping/recreation, etc.

I see absolutely no purpose to keeping a second compact car. I also see no purpose in buying the new car if you are too silly to be willing to drive it on short trips like errands.

"I've always figured I'd run the car to the ground before getting rid of it" No one is forcing you to buy a new car yet. I have a 2000 Honda CR-V and still love it.

You won't get much for trade-in or if you sell it - virtually no value. However, it will still cost $$$$ to insure… Years back when I lived in PA, a single driver could buy a policy that gave them liability coverage on multiple cars you owned if you were the only driver (ie, you couldn't drive 2 at once - but no other drivers covered) then you could separately buy a set value collision/comp with a different insured value for each car. Don't know if such policies are still available - probably not except for "collectors cars".

A) The $500 trade-in the dealer is offering means you're getting nothing for the car at all. They're just hiding that fact in a higher price for the new car. Anyone good at negotiating could have haggled them down that $500 without trading the car, which is undoubtedly worth more on the private market if you at least fix the most critical issue, which is probably the radiator (it might just be a bad or loose hose). When buying a new vehicle, NEVER bring up the subject of a trade-in until AFTER the best deal you can get has already been negotiated and accepted. You may have already squandered your "bargaining chip" at that particular dealership. I wouldn't trade the car. I'd sooner walk out and go to another dealer if they're not willing to drop the price another $500 without the trade-in.

b) While there's some logic in keeping the beater to avoid putting excess wear and tear on the new vehicle, and I can understand your emotional attachment (I still have my first car, and I've kept it as a 2nd or 3rd vehicle for over 30 years), you have to factor in the price of registering and insuring it, as well as the extra space it takes up, along with the cost of keeping it running. That's your call.

c) If you'd do at least some of the maintenance work yourself, you could cut that repair cost by a couple hundred bucks, if not more. The savings might be enough to cover insurance and registration for another year.

d) If the Civic is still reasonably reliable and isn't a complete rust bucket, there's a ready market for it because there. Always someone looking for a cheap beater that runs. Heck, you could even part it out and get more than the $500 as a trade in, if you're willing to go to the trouble.

IF you get a new primary vehicle, you can then do the work yourself and repair the Honda, learning AND saving $$$$. I have three cars and my Girlfriend has two, and I'm always taking something apart…

Only if no-one else in your family has a spare car you could use if the primary one broke down. Otherwise you're wasting insurance and registration/annual road tax and also facing a steepening repair bill for it. Probably getting rubbish MPG, compared to a modern car, from it too.
Keep it for a while, fix it and then sell it privately for 995.

There's no right or wrong answer. In the end it will be cheaper to own just one vehicle, or no vehicle if your circumstances allow. With two vehicles you have to pay insurance, registration, maintenance and possibly emissions testing on both vehicles. But short trips are hard on gasoline powered vehicles and if one breaks down you have another, so there's some logic in owning two. It will cost more but has it's benefits.

Personally if I were to keep two vehicles I'd make one of those a cheap used electric car, like an electric Smart Car for under $5,000. In most places electricity will cost a third of what you're paying for gasoline or diesel and maintenance will be nearly zero. Then fix those tie rods and keep the Honda for those occasional long trips when the EV won't make it. After all, gasoline cars get better fuel mileage and live longer if you use them for long distance travel.

Probably does not make sense, but keep it anyway

I would keep it. Those are normal repairs for a car that old. You can space out the repairs over a few months.
As for insurance, just having liability on the old car will not really increase your insurance costs.
And one factor is by putting fewer miles on the newer vehicle it depreciates less. So if you get a year or so out of it, you have your money back, and you can always sell it for say $1000? And if it has a terminal failure in future, then you just junk it.
Also since it is an older vehicle maybe you can find a shop that will give you a break on parts or let you buy your own parts.
Not all mechanics will do this, only a smaller independent shop. I have two cars and one thing I have found is I tell the mechanic, I will bring you parts needed, so you don't have to deal with that, and I can leave the car for week and you can work on it when you have time.
Regarding the rack, I would have a flush done before the new rack is put in. This is a bit of overkill but you need to flush it to get any debris out (the seals can break down and put debris in the fluid), and put spec fluid in there, ie you do not want to mix types and brands of fluid. So when you get the new rack the system is clean and any residual fluid is spec fluid. They should fully drain the system before putting in the new rack.

I'm on my 3rd Honda; they last forever.
Fix the rods & steering & use it for local trips.